On derivations of some classes of Leibniz algebras

In the paper we describe the derivations of complex $n$-dimensional naturally graded filiform Leibniz algebras $NGF_1, NGF_2\ \text{and} \ \ NGF_3.$ We show that the dimension of the derivation algebras of $NGF_1$ and $NGF_2$ equals $n+1$ and $n+2,$ respectively, while the dimension of the derivation algebra of $NGF_3$ is equal to $2n-1.$ The second part of the paper deals with the description of the derivations of complex $n$-dimensional filiform non Lie Leibniz algebras, obtained from naturally graded non Lie filiform Leibniz algebras. It is well known that this class is split into two classes denoted by $FLb_n$ and $SLb_n.$ Here we found that for $L\in FLb_n$ we have $n-1 \leq dim\ Der(L)\leq n+1$ and for algebras $L$ from $SLb_n$ the inequality $n-1\leq dim\ Der(L) \leq n+2$ holds true.


Introduction
A graded algebra is an algebra endowed with a gradation which is compatible with the algebra bracket. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra. Lie algebra sl 2 of trace-free 2 × 2 matrices is graded by the generators: the decomposition sl 2 = g −1 ⊕ g 0 ⊕ g 1 presents sl 2 as a graded Lie algebra. It is well-known that the natural gradation of nilpotent Lie and Leibniz algebras is very helpful in investigation of their structural properties. This technique is more effective when the length of the natural gradation is sufficiently large. The case when it is maximal the algebra is called filiform. For applications of this technique, for instance, see [9] and Goze et al. [3] (for Lie algebras) and [1] (for Leibniz algebras) case. In [9] Vergne introduced the concept of naturally graded filiform Lie algebras as those admitting a gradation associated with the lower central series. In that paper, she also classified them, up to isomorphism. Apart from that, several authors have studied algebras which admit a connected gradation of maximal length, this is, whose length is exactly the dimension of the algebra. So, Khakimdjanov started this study in [4], Reyes, in [2], continued this research by giving an induction classification method and finally, Millionschikov in [7] gave the full list of these algebras (over an arbitrary field of zero characteristic).
Recall that an algebra L over a field K is called Leibniz algebra if it satisfies the following Leibniz identity: where [·, ·] denotes the multiplication in L (First the Leibniz algebras has been introduced in [6]). It is not difficult to see that the class of Leibniz algebras is "nonantisymmetric" generalization of the class of Lie algebras. In this paper we are dealing with the derivations of some classes of complex Leibniz algebras.
The outline of the paper is as follows. Section 2 contains preliminary results on Leibniz algebras which we will use in the paper. The main results of the paper are in Section 3. The first part of this section deals with the description of derivations of naturally graded Leibniz algebras. In the second part (Section 3.2) we study derivations of filiform Leibniz algebras arising from naturally graded non Lie filiform Leibniz algebras. It is known that the last is split into two disjoint subclasses [1]. In the paper we denote these classes by FLb n , and S Lb n . We show that according to dimensions of the derivation algebras each class is split into subclasses as follows: where F i and S j are subclasses of FLb n and S Lb n , respectively, with the derivation algebras' dimensions i and j.
Further all algebras considered are over the field of complex numbers C and omitted products of basis vectors are supposed to be zero.

Preliminaries
This section contains definitions and results which will be needed throughout the paper.
Let L be a Leibniz algebra. We put: Definition 1. A Leibniz algebra L is said to be nilpotent if there exists s ∈ N such that Obviously, a filiform Leibniz algebra is nilpotent.
The set of all derivations of an algebra L is denoted by Der(L). By Lb n we denote the set of all n-dimensional filiform Leibniz algebras, appearing from naturally graded non Lie filiform Leibniz algebras. For Lie algebras the study of derivations has been initiated in [5]. The derivations of naturally graded filiform Leibniz algebras were first considered by Omirov in [8]. In the following theorem we declare the results of the papers [1], [9]. Theorem 1. Any complex n-dimensional naturally graded filiform Leibniz algebra is isomorphic to one of the following pairwise non isomorphic algebras: where α ∈ {0, 1} for even n and α = 0 for odd n.

Derivations of graded Leibniz algebras.
In this section we study the derivations of NGF i , i = 1, 2, 3. In each case we give a basis of the derivation algebra. Let d be represented by a matrix D = (d l k ), k,l=1,2,3,...,n, on the basis {e 1 , e 2 , ..., e n }. We describe the matrix D.
Proof. Let us start from NGF 1 . We take d(e j ) = Therefore, Hence, Comparing (3.1) and (3.2) we obtain According to the table of multiplication of NGF 1 , one has [e 3 , e 1 ] = e 4 . Thus Therefore, For k ≥ 5 one can find Indeed, it is true for k = 4. Suppose that it is true for k and show that it is the case for Hence, we get In fact, e n = [e n−1 , e 1 ] therefore, We substitute k by n − 1 in (3.5) and obtain d(e n−1 ) = ((n − 2)d 1 The matrix of d on the basis {e 1 , e 2 , e 3 , ..., e n } has the following form: Consider the following system of vectors: where E i j is the matrix with zero entries except for the element a i j = 1. It is easy to see that the set {v 1 , v 2 , v 3 , ...v n+1 } presents a basis of Der (NGF 1 ), therefore, dim Der (NGF 1 ) = n + 1.
Next, we describe the derivation algebra of If one uses [e 2 , e 1 ] = 0 then Therefore, Because of [e 3 , e 1 ] = e 4 we find Similarly, Then the matrix of d has the form From the view of D it is easy to conclude that dim Der(NGF 2 ) = n + 2.
Let us now consider the derivation algebra of NGF 3 . We take d(e j ) = Therefore,  where Thus the dimension of Der(NGF 3 ) is 2n − 1.

Derivations of filiform Leibniz algebras. Now we study the derivations of classes from Theorem 2.
Theorem 4. The dimensions of the derivation algebras of FLb n are equal to n − 1, n or n + 1.
Proof. Depending on constraints for the structure constants α 4 , α 5 , ..., α n−1 and θ we have the following distribution for dimensions of the derivation algebras of elements from FLb n : Comparing the last two expressions for d(e 3 ) we obtain Similarly, We substitute k by n − 2 in (3.12), to obtain Comparing (3.13) and (3.14) we obtain The matrix of d has the form D = (d l k ) k,l=1,2,3,...n , where Hence, in this case the dimension of Der(L) for L ∈ FLb n is n.
Now we describe the derivation algebra of elements from S Lb n .
Theorem 5. The dimensions of the derivation algebras for elements of S Lb n vary between n − 1 and n + 2.
Proof. Similarly to the case of FLb n for the class S Lb n we have the distribution for dimension of derivation algebra as follows.
Therefore we obtain   (3.25) The dimension of the derivation algebra of L ∈ S Lb n is n − 1.
The other cases are treated similarly. And at last, if L ∈ S Lb n with γ = 0 and α i = 0, for i = 4, 5, .., n−1. Then dimension of the derivation algebra of L is n + 2, which is immediate from the case of NGF 2 .